\[ y'(x)=\frac {(a-1) (a+1) x}{a^2 F\left (-\frac {1}{2} a^2 x^2+\frac {x^2}{2}+\frac {y(x)^2}{2}\right )-F\left (-\frac {1}{2} a^2 x^2+\frac {x^2}{2}+\frac {y(x)^2}{2}\right )+y(x)} \] ✓ Mathematica : cpu = 0.304814 (sec), leaf count = 177
DSolve[Derivative[1][y][x] == ((-1 + a)*(1 + a)*x)/(-F[x^2/2 - (a^2*x^2)/2 + y[x]^2/2] + a^2*F[x^2/2 - (a^2*x^2)/2 + y[x]^2/2] + y[x]),y[x],x]
\[\text {Solve}\left [\int _1^{y(x)}\left (\frac {K[2]}{(a-1) (a+1) F\left (-\frac {1}{2} a^2 x^2+\frac {x^2}{2}+\frac {K[2]^2}{2}\right )}-\int _1^x\frac {K[1] K[2] F'\left (-\frac {1}{2} a^2 K[1]^2+\frac {K[1]^2}{2}+\frac {K[2]^2}{2}\right )}{F\left (-\frac {1}{2} a^2 K[1]^2+\frac {K[1]^2}{2}+\frac {K[2]^2}{2}\right )^2}dK[1]+1\right )dK[2]+\int _1^x-\frac {K[1]}{F\left (-\frac {1}{2} a^2 K[1]^2+\frac {K[1]^2}{2}+\frac {y(x)^2}{2}\right )}dK[1]=c_1,y(x)\right ]\] ✓ Maple : cpu = 0.306 (sec), leaf count = 60
dsolve(diff(y(x),x) = x*(a-1)*(a+1)/(y(x)+F(1/2*y(x)^2-1/2*a^2*x^2+1/2*x^2)*a^2-F(1/2*y(x)^2-1/2*a^2*x^2+1/2*x^2)),y(x))
\[\frac {y \left (x \right )}{\left (a -1\right ) \left (a +1\right )}+\frac {\int _{}^{-a^{2} x^{2}+x^{2}+y \left (x \right )^{2}}\frac {1}{F \left (\frac {\textit {\_a}}{2}\right )}d \textit {\_a}}{2 a^{4}-4 a^{2}+2}-c_{1} = 0\]